Generalized Definitions

The following discussion uses various concepts, which are first defined. The general definitions given here hold for simple reactions only, and are subsequently elaborated further to cover different situations, shapes, conditions etc. The three most important concepts are  

The effectiveness factor r of a reaction is defined as the ratio of the amount of a component (participating in that reaction) converted inside a catalyst pellet to the amount that would have been converted if the conditions on the outer surface were to prevail everywhere inside the catalyst pellet. Hence  

The zeroth Aris number Ana is defined as the number which becomes equivalent to 1/ if if the effectiveness factor 17 goes to zero. Hence  

From this it follows that An0 is defined only if 77 can get infinitely close to zero, which does not always hold, as we will see. 

---

Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

The are many ways to define the rate of a chemical reaction. The most general definition uses the rate of change of a themiodynamic state function. Following the second law of themiodynamics, for example, the change of entropy S with time t would be an appropriate definition under reaction conditions at constant energy U and volume V ... 

Now let us use the wavepackets just discussed to extract the physically measurable infomration about our problem, namely, the probabilities of reflection and transmission. As long as the wavepackets do not spread much during the collision, these probabilities are given by the general definition ... 

We have seen that a base can be defined as combining with a proton and, therefore, requires at least one lone pair of electrons. A more general definition of acids and bases, due to G. N. Lewis, describes a base as any species (atom, ion or molecule) which can donate an electron pair, and an acid as any species which can accept an electron pair— more simply, a base is an electron-pair donor, an acid an electron-pair acceptor. Some examples of Lewis acids and bases are ... 

Chemoinfonnatics is a faiily new name for a discipline that, as we shall soon see, has been around for quite a while. Different people sometimes give rather divergent definitions of chemoinfoimatics. Before we discuss these different viewpoints, let us, for the time being, accept a rather broad and general definition ... 

The units of concentration most frequently encountered in analytical chemistry are molarity, weight percent, volume percent, weight-to-volume percent, parts per million, and parts per billion. By recognizing the general definition of concentration given in equation 2.1, it is easy to convert between concentration units. 

Equation (2.39) is the weight average molecular weight as defined in Sec. 1.8. It is important to note that this result. My = M y, applies only in the case of nonentangled chains where 17 is directly proportional to M. A more general definition of My for the case where 17 a is... 

For each essential component of the invention, have the foUowing been identified its function a general definition of the component a specific Hst of materials which may be used for each component what is the preferred material limits, boundaries, ranges at which the material is useful preferred limits, boundaries, ranges ... 

Chemurgy is defined as that branch of appHed chemistry devoted to industrial utilization of organic raw materials, especially from farm products. A more modem and general definition for chemurgy is the use of renewable resources particularly biomass, usually plant or microbial material, for materials and energy (see Fuels frombiomass Fuels fromwaste). 

High voltage test techniques - General definitions and test requirement 2071-1/199.3 BS 92.3-1/1990... 

The general definition of a condensation reaction is a one that involves product formation by expulsion of water (or other small molecule) as a by-product. By this definition, activation and methylolation are also condensations. In more precise terms the chain-building process should be described as a condensation polymerization, however, in the jargon of the phenolics industry, the term condensation is usually reserved for the chain-building process. This terminology is not necessarily observed in the literature [88]. Many literature reports correctly refer to methylolation as a condensation reaction. The molecular weight development of the phenol alcohol adducts may also be classified as a step-polymerization. 

Elevated Flares See Flares for a general definition. The elevated flare, by the use of steam injection and effective tip design, operates as a smokeless combustion device. Flaring generally is of low luminosity up to about 20 % of maximum flaring load. Steam injection tends to introduce a source of noise to the operation, and a compromise between smoke elimination and noise is usually necessary. When adequately elevated (by means of a stack) this type of flare displays the best dispersion characteristics for malodorous and toxic combustion products. Visual and noise pollution often creates nuisance problems. Capital and operating costs tend to be high, and an appreciable plant area can be rendered unavailable for plant operations and equipment because of excessive radiant heat. 

We recognize that 0id majority merely a special case of the generalized threshold rule defined in equation 5.121. In particulm-, majority may be defined using that more general definition by setting... 

We will consider more general definitions of acids and bases in Chapter 13 (Bransted-Lowry) and Chapter 15 (Lewis). 

Having considered all the possible cases, we are now in a position to give a more general definition of hydrolysis. Hydrolysis is the interaction between an ion (or ions) of a salt and water with the production of (a) a weak acid or a weak base, or (b) of both a weak acid and a weak base. 

A general definition for a high-tempera-ture plastic is one having a thermal value in terms of ASTM D 648 and UL 746B higher than 149°C (300°F). There are numerous plastics that are both processable and have... 

When chemists see a pattern in the reactions of certain substances, they formulate a definition of a class of substance that captures them all. The reactions of the substances we call acids and bases are an excellent illustration of this approach. The pattern in these reactions was first identified in aqueous solutions, and led to the Arrhenius definitions of acids and bases (Section J). However, chemists discovered that similar reactions take place in nonaqueous solutions and even in the absence of solvent. The original definitions had to be replaced by more general definitions that encompassed this new knowledge. 

EU legislation provides the following general definition of a medical device ... 

Here in Chapter 1 we make the additional assumptions that the fluid has constant density, that the cross-sectional area of the tube is constant, and that the walls of the tube are impenetrable (i.e., no transpiration through the walls), but these assumptions are not required in the general definition of piston flow. In the general case, it is possible for u, temperature, and pressure to vary as a function of z. The axis of the tube need not be straight. Helically coiled tubes sometimes approximate piston flow more closely than straight tubes. Reactors with square or triangular cross sections are occasionally used. However, in most of this book, we will assume that PFRs are circular tubes of length L and constant radius R. 

Equation (4.10) uses the general definition of fraction unreacted in a flow system. It is moles out divided by moles in. The corresponding, general definition of conversion is... 

The double integral in Equation (8.4) is a fairly general definition of the mixing-cup average. It is applicable to arbitrary velocity profiles and noncircular cross sections but does assume straight streamlines of equal length. Treatment of curved streamlines requires a precise and possibly artificial definition of the system boundaries. See Nauman and Buffham. ... 

The p-stability condition. A more general definition of stability with respect to the initial data became rather urgent and extremely important. 

Cady, H. P. Elsey, H. M. (1928). A general definition of acids, bases and salts. Journal of Chemical Education, 5, 1425-8. 

A general definition of log P and log D, in its simplest form, can be given as the logarithm of the ratio (P or D) of the concentration of species of interest (the drug in a pharmaceutical context) in each phase, assuming the phases are immiscible and well separated prior to analysis. P is defined as the partition coefficient, whereas D is the distribution coefficient. However, the simplest form does not reveal some of the intricacies of the determination and use of these parameters, and further explanation is necessary. 

Peedc-to-peak resolution in SEC can be calculated by the ratio of peak separation at the peak maxiaut to the sum of the baseline peak widths. This general definition of resolution is less useful in SEC, where a measure of the ability of the column to separate solutes of different molecular weight is required. For this purposes, we define a new term, the specific resolution factor, R, which relates peak resolution to sample molecular weight, assuming all measurements are made within the linear region of the molecular weight calibration curve, equation (4.41)... 

Any text on acids and bases would not be deemed complete if mention were not made of the extended definition of acids and bases that is embodied in the Lowry-Bronsted theory. The theory basically proposed a more general definition of acids and bases to overpower the limitations of the theory arising from the Arrhenius concept. 

From both a theoretical and practical view, it is ideal to use Bayesian Decision Theory because it represents an optimal classifier. From a theoretical perspective, Bayesian Decision Theory offers a general definition of the pattern recognition problem and, with appropriate assumptions, it can be shown to be the basis of many of the so-called non-PDF approaches. In practice, however, it is typically treated as a separate method because it places strong data availability requirements for direct use compared to other approaches. 

The constant matrices i- act as projection operators onto the different eigenspaces. They are given in Ref. 38. The solution Eq. (30) is entirely analogous to Eq. (20) in the white noise case. To obtain a trajectory that remains in the vicinity of the barrier for all times, we again have to set caj = 0 and identify Eq. (31) as the TS trajectory. It satisfies the condition of the general definition in that it provides, at fixed time, a random ensemble of trajectories that is stationary in time, and at fixed noise sequence a a trajectory that spends most of its time close to the barrier. 

In particular, the TS trajectory remains bounded for all times, which satisfies the general definition. The constants c and c in Eq. (39) depend on the specific choice of the TS trajectory. Because the saddle point of the autonomous system becomes a fixed point for large positive and negative times, one might envision an ideal choice to be one that allows the TS trajectory to come to rest at the saddle point both in the distant future and in the remote past. However, this is impossible in general because the driving force will transfer energy into or out of the bath modes in such a way that... 

---